Problem: Simplify the following expression: $ r = \dfrac{4}{5} + \dfrac{t + 7}{-9t + 7} $
In order to add expressions, they must have a common denominator. Multiply the first expression by $\dfrac{-9t + 7}{-9t + 7}$ $ \dfrac{4}{5} \times \dfrac{-9t + 7}{-9t + 7} = \dfrac{-36t + 28}{-45t + 35} $ Multiply the second expression by $\dfrac{5}{5}$ $ \dfrac{t + 7}{-9t + 7} \times \dfrac{5}{5} = \dfrac{5t + 35}{-45t + 35} $ Therefore $ r = \dfrac{-36t + 28}{-45t + 35} + \dfrac{5t + 35}{-45t + 35} $ Now the expressions have the same denominator we can simply add the numerators: $r = \dfrac{-36t + 28 + 5t + 35}{-45t + 35} $ $r = \dfrac{-31t + 63}{-45t + 35}$ Simplify the expression by dividing the numerator and denominator by -1: $r = \dfrac{31t - 63}{45t - 35}$